There Are Only Two!
There exist two ordered triples for which represents a pair of identical straight lines in the -plane.
If these triples are and , then find the value of .
The answer is 2.
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1 solution
Suppose the identical lines take the form m x + n y + p = 0 . Then their product is ( m x + n y + p ) ( m x + n y + p ) = 0 ⟶ m 2 x 2 + 2 m n x y + n 2 y 2 + 2 m p x + 2 n p y + p 2 = 0 .
In order for this equation to fit the form of the given equation, we need m 2 = 4 , 2 m n = − 4 , and p 2 = 1 .
Case I) m = 2 , n = − 1 , p = 1 : In this case, the product of the lines is ( 2 x − y + 1 ) ( 2 x − y + 1 ) = 4 x 2 − 4 x y + y 2 + 4 x − 2 y + 1 = 0 , meaning a = 1 , b = 4 , and c = − 2 .
Case II) m = 2 , n = − 1 , p = − 1 : In this case, the product of the lines is ( 2 x − y − 1 ) ( 2 x − y − 1 ) = 4 x 2 − 4 x y + y 2 − 4 x + 2 y + 1 = 0 , meaning a = 1 , b = − 4 , and c = 2 .
Case III) m = − 2 , n = 1 , p = 1 : This is a duplicate of Case I, since each equation is being multiplied by − 1 . Same as if we had let p = − 1 .
Adding the values of a , b , and c obtained so far gives us 1 + 4 − 2 + 1 − 4 + 2 = 2 .